# To further justify the Cofunction Theorem, use your calculator to find a value for the given pair of trigonometric functions. In each case, the trigonometric functions are cofunctions of one another, and the angles are complementary angles. Round your answers to four places past the decimal point. sec 6.7^@, cosec 83.3^@

Question
Trigonometric Functions
To further justify the Cofunction Theorem, use your calculator to find a value for the given pair of trigonometric functions. In each case, the trigonometric functions are cofunctions of one another, and the angles are complementary angles. Round your answers to four places past the decimal point.
$$\displaystyle{{\sec{{6.7}}}^{\circ},}{\cos{{e}}}{c}{83.3}^{\circ}$$

2021-02-26
Use cofunction of the given trigonometric function,
$$\displaystyle{\sec{{x}}}={\cos{{e}}}{c}{\left({90}^{\circ}−{x}\right)}$$
Given,
$$\displaystyle{{\sec{{6.7}}}^{\circ}}$$
Here $$\displaystyle{x}={6.7}^{\circ}$$
$$\displaystyle{{\sec{{6.7}}}^{\circ}=}{\cos{{e}}}{c}{\left({90}^{\circ}−{6.7}^{\circ}\right)}$$
$$\displaystyle={\cos{{e}}}{c}{\left({83.3}^{\circ}\right)}$$
$$\displaystyle={1.006876}$$
$$\displaystyle\approx{1.0069}$$
Therefore $$\displaystyle{{\sec{{6.7}}}^{\circ}=}{1.0069}$$
Now use cofunction of the given trigonometric function,
$$\displaystyle{\cos{{e}}}{c}{x}={\sec{{\left({90}^{\circ}−{x}\right)}}}$$
Given,
$$\displaystyle{\cos{{e}}}{c}{\left({83.3}^{\circ}\right)}$$
Here $$\displaystyle{x}={83.3}^{\circ}$$
$$\displaystyle{\cos{{e}}}{c}{83.3}^{\circ}={\sec{{\left({90}^{\circ}−{83.3}^{\circ}\right)}}}$$
$$\displaystyle={{\sec{{6.7}}}^{\circ}}$$
$$\displaystyle={1.006876}$$
$$\displaystyle\approx{1.0069}$$
Therefore, $$\displaystyle{\cos{{e}}}{c}{83.3}^{\circ}={1.0069}$$
We clearly see that trigonometric functions are cofunction of one another.
Sum of the angles of two cofunctions is $$\displaystyle{90}^{\circ}$$ that is the angles are complementary angles.
Therefore the pair of a trigonometric function is,
$$\displaystyle{\left({{\sec{{6.7}}}^{\circ},}{\cos{{e}}}{c}{83.3}^{\circ}\right)}={\left({1.0069},{1.0069}\right)}$$

### Relevant Questions

A new thermostat has been engineered for the frozen food cases in large supermarkets. Both the old and new thermostats hold temperatures at an average of $$25^{\circ}F$$. However, it is hoped that the new thermostat might be more dependable in the sense that it will hold temperatures closer to $$25^{\circ}F$$. One frozen food case was equipped with the new thermostat, and a random sample of 21 temperature readings gave a sample variance of 5.1. Another similar frozen food case was equipped with the old thermostat, and a random sample of 19 temperature readings gave a sample variance of 12.8. Test the claim that the population variance of the old thermostat temperature readings is larger than that for the new thermostat. Use a $$5\%$$ level of significance. How could your test conclusion relate to the question regarding the dependability of the temperature readings? (Let population 1 refer to data from the old thermostat.)
(a) What is the level of significance?
State the null and alternate hypotheses.
$$H0:?_{1}^{2}=?_{2}^{2},H1:?_{1}^{2}>?_{2}^{2}H0:?_{1}^{2}=?_{2}^{2},H1:?_{1}^{2}\neq?_{2}^{2}H0:?_{1}^{2}=?_{2}^{2},H1:?_{1}^{2}?_{2}^{2},H1:?_{1}^{2}=?_{2}^{2}$$
(b) Find the value of the sample F statistic. (Round your answer to two decimal places.)
What are the degrees of freedom?
$$df_{N} = ?$$
$$df_{D} = ?$$
What assumptions are you making about the original distribution?
The populations follow independent normal distributions. We have random samples from each population.The populations follow dependent normal distributions. We have random samples from each population.The populations follow independent normal distributions.The populations follow independent chi-square distributions. We have random samples from each population.
(c) Find or estimate the P-value of the sample test statistic. (Round your answer to four decimal places.)
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?
At the ? = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.At the ? = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the ? = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.At the ? = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.
(e) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is sufficient evidence that the population variance is larger in the old thermostat temperature readings.Fail to reject the null hypothesis, there is sufficient evidence that the population variance is larger in the old thermostat temperature readings. Fail to reject the null hypothesis, there is insufficient evidence that the population variance is larger in the old thermostat temperature readings.Reject the null hypothesis, there is insufficient evidence that the population variance is larger in the old thermostat temperature readings.
Use a calculator to find the value of the trigonometric function csc 1 to four decimal places.
Use a calculator to find the value of the trigonometric function $$\displaystyle{\sin{{\left(\frac{{{3}\pi}}{{10}}\right)}}}$$ to four decimal places.
Would you rather spend more federal taxes on art? Of a random sample of $$n_{1} = 86$$ politically conservative voters, $$r_{1} = 18$$ responded yes. Another random sample of $$n_{2} = 85$$ politically moderate voters showed that $$r_{2} = 21$$ responded yes. Does this information indicate that the population proportion of conservative voters inclined to spend more federal tax money on funding the arts is less than the proportion of moderate voters so inclined? Use $$\alpha = 0.05.$$ (a) State the null and alternate hypotheses. $$H_0:p_{1} = p_{2}, H_{1}:p_{1} > p_2$$
$$H_0:p_{1} = p_{2}, H_{1}:p_{1} < p_2$$
$$H_0:p_{1} = p_{2}, H_{1}:p_{1} \neq p_2$$
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A random sample of $$\displaystyle{n}_{{1}}={16}$$ communities in western Kansas gave the following information for people under 25 years of age.
$$\displaystyle{X}_{{1}}:$$ Rate of hay fever per 1000 population for people under 25
$$\begin{array}{|c|c|} \hline 97 & 91 & 121 & 129 & 94 & 123 & 112 &93\\ \hline 125 & 95 & 125 & 117 & 97 & 122 & 127 & 88 \\ \hline \end{array}$$
A random sample of $$\displaystyle{n}_{{2}}={14}$$ regions in western Kansas gave the following information for people over 50 years old.
$$\displaystyle{X}_{{2}}:$$ Rate of hay fever per 1000 population for people over 50
$$\begin{array}{|c|c|} \hline 94 & 109 & 99 & 95 & 113 & 88 & 110\\ \hline 79 & 115 & 100 & 89 & 114 & 85 & 96\\ \hline \end{array}$$
(i) Use a calculator to calculate $$\displaystyle\overline{{x}}_{{1}},{s}_{{1}},\overline{{x}}_{{2}},{\quad\text{and}\quad}{s}_{{2}}.$$ (Round your answers to two decimal places.)
(ii) Assume that the hay fever rate in each age group has an approximately normal distribution. Do the data indicate that the age group over 50 has a lower rate of hay fever? Use $$\displaystyle\alpha={0.05}.$$
(a) What is the level of significance?
State the null and alternate hypotheses.
$$\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}<\mu_{{2}}$$
$$\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}>\mu_{{2}}$$
$$\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}\ne\mu_{{2}}$$
$$\displaystyle{H}_{{0}}:\mu_{{1}}>\mu_{{2}},{H}_{{1}}:\mu_{{1}}=\mu_{{12}}$$
(b) What sampling distribution will you use? What assumptions are you making?
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations,
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations,
The Student's t. We assume that both population distributions are approximately normal with known standard deviations,
What is the value of the sample test statistic? (Test the difference $$\displaystyle\mu_{{1}}-\mu_{{2}}$$. Round your answer to three decimalplaces.)
What is the value of the sample test statistic? (Test the difference $$\displaystyle\mu_{{1}}-\mu_{{2}}$$. Round your answer to three decimal places.)
(c) Find (or estimate) the P-value.
P-value $$\displaystyle>{0.250}$$
$$\displaystyle{0.125}<{P}-\text{value}<{0},{250}$$
$$\displaystyle{0},{050}<{P}-\text{value}<{0},{125}$$
$$\displaystyle{0},{025}<{P}-\text{value}<{0},{050}$$
$$\displaystyle{0},{005}<{P}-\text{value}<{0},{025}$$
P-value $$\displaystyle<{0.005}$$
Sketch the sampling distribution and show the area corresponding to the P-value.
P.vaiue Pevgiue
P-value f P-value
What is the optimal time for a scuba diver to be on the bottom of the ocean? That depends on the depth of the dive. The U.S. Navy has done a lot of research on this topic. The Navy defines the "optimal time" to be the time at each depth for the best balance between length of work period and decompression time after surfacing. Let $$\displaystyle{x}=$$ depth of dive in meters, and let $$\displaystyle{y}=$$ optimal time in hours. A random sample of divers gave the following data.
$$\begin{array}{|c|c|} \hline x & 13.1 & 23.3 & 31.2 & 38.3 & 51.3 &20.5 & 22.7 \\ \hline y & 2.78 & 2.18 & 1.48 & 1.03 & 0.75 & 2.38 & 2.20 \\ \hline \end{array}$$
(a)
Find $$\displaystyleΣ{x},Σ{y},Σ{x}^{2},Σ{y}^{2},Σ{x}{y},{\quad\text{and}\quad}{r}$$. (Round r to three decimal places.)
$$\displaystyleΣ{x}=$$
$$\displaystyleΣ{y}=$$
$$\displaystyleΣ{x}^{2}=$$
$$\displaystyleΣ{y}^{2}=$$
$$\displaystyleΣ{x}{y}=$$
$$\displaystyle{r}=$$
(b)
Use a $$1\%$$ level of significance to test the claim that $$\displaystyle\rho<{0}$$. (Round your answers to two decimal places.)
$$\displaystyle{t}=$$
critical $$\displaystyle{t}=$$
Conclusion
Reject the null hypothesis. There is sufficient evidence that $$\displaystyle\rho<{0}$$.Reject the null hypothesis. There is insufficient evidence that $$\displaystyle\rho<{0}$$.
Fail to reject the null hypothesis. There is sufficient evidence that $$\displaystyle\rho<{0}$$.Fail to reject the null hypothesis. There is insufficient evidence that $$\displaystyle\rho<{0}.$$
(c)
Find $$\displaystyle{S}_{{e}},{a},{\quad\text{and}\quad}{b}$$. (Round your answers to four decimal places.)
$$\displaystyle{S}_{{e}}=$$
$$\displaystyle{a}=$$
$$\displaystyle{b}=$$
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A fair 7 -sided die with the numbers 1 trough 7 is rolled five times. Express each of your answers as a decimal rounded to 3 decimal places.
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